I am trying to work with Kou's double exponential Jump-diffusion model and simulate a price path in a programming language.
So the dynamics of the asset price in Kou's model follow: \begin{equation} \frac{dS(t)}{S(t-)}=\mu dt+\sigma dW(t)+d(\sum_{i=1}^{N(t)}(V_i-1)) \end{equation}
where W(t) is a standard Brownian motion, N(t) is a Poisson process with rate λ , and {Vi} is a sequence of independent identically distributed (i.i.d.) non negative random variables such that Y=log(V) has an asymmetric double exponential distribution with the density: \begin{equation} f_Y(y)=p.\eta_1 e^{-\eta_{1}y}\upharpoonleft_{y\geq 0}+q.\eta_2 e^{\eta_2 y} \upharpoonleft_{y<0},\eta_{1}>1,\eta_{2}>0 \end{equation}
Solving this SDE gives: \begin{equation} S(t)=S(0)\exp\{(\mu- \frac{1}{2}\sigma^2)t+\sigma W(t)\} \prod_{i=1}^{N(t)}V_i \end{equation}
I generate the Yi-s in a simulation program via the asymmetric double exponential distribution. So let's say I have generated the following four jumps: \begin{equation} \{12.8277,-14.4736,7.287,-10.1267\} \end{equation}
EDIT: I simulate these values with the following Matlab code:
y=binornd(1,p,N,1); %1 = upwards jump, 0 = downwards jump
Y=y.*exprnd(e1,N,1)-(1-y).*exprnd(e2,N,1);
Now the part which I do not get is the following. Because Y = log(V), the Vi-s in the price equation are: \begin{equation} V_i = e^{Y_i} \end{equation} right?
So when the first jump occurs at time t1, I am adding the jump part in the price equation (the multiplication with Vi). To do so, I take the exponential of 12.8277, but then the stock price explodes (because exp(12.8277)>372).
I think I am mixing things up with the exponential in the equation, because multiplying with the exponential of the generated Yi-s leads to incorrect stock prices.
Could someone explain to me the part which I am interpreting wrong?